Chapter 3.7, Problem 20E

To determine

To find: equations of vertical and horizontal asymptotes, if any, of the given function.

Answer to Problem 20E

  $\begin{array}{l}x=2\left[\text{Vertical asymptote}\right]\\ y=0\left[\text{Horizontal asymptote}\right]\end{array}$

Explanation of Solution

Given information:

  $y=\frac{x^3}{{\left(x-2\right)}^4}$

Calculation:

Put the denominator equal to 0 and evaluate x as follows:

  $\begin{array}{c}{\left(x-2\right)}^4=0\\ x-2=0\\ x=2\end{array}$

The vertical asymptote is $x=2$ .

Look at the degree of numerator and denominator to find the horizontal asymptotes as follows:

  $f\left(x\right)=\frac{a{x}^n}{b{x}^m}$

Case	Horizontal Asymptote
$n<m$	$y=0$
$n=m$	$y=\frac{a}{b}$
$n>m$	No horizontal asymptote

Since the degree of numerator and denominator in the given function is 3 and 4 respectively. Thus, the horizontal asymptote is $y=0$ .